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Can get very large. Some examples are 5 6 7 8 9 n 0 1 2 3 4 n! 1 1 2 6 24 120 720 5040 40320 362880 To approximate the value for large n, a refined Stirlings’s approximation can be used . n! = (2π)1/2 nn+1/2 e−n+1/(12n) . Next, consider the number of different subsets of size r from n different items, with the order of the items within the subset not distinguished. 3 Combination rule. The number of different subsets of size r chosen from a set of n different items is (n)r n = r r! which is called the number of combinations of n things taken r at a time.

Calculate the Walsh sum median. 16. Calculate the sample mean. 17. 13). 18. 4). 19. 13). 20. Prove D ≤ R/2. 1 The Sample Space Consider a random experiment which has a variety of possible outcomes. Let us denote the set of possible outcomes, called the sample space, by S = {e1 , e2 , e3 , . }. If the outcome ei belongs to the set S we write ei ∈ S. If the outcome does not belong to the set S we write e ∈ S. Ws say the sample space is discrete if either there is a finite number of possible outcomes S = {ei : i = 1, 2, .

N. Then if we denote the number of outcomes in the set A by ||A|| we have P (A) = i: ei ∈A 1 1 = N N 1= i: ei ∈A ||A|| . ||S|| For such probabilities, it is necessary to count the number of outcomes ||A|| and ||S|| = N. We give some standard counting rules. 1 Product rule. If one operation can be done in n1 ways and the second operation can be done in n2 ways independently of the first operation then the pair of operations can be done in n1 × n2 ways. If we have K operations, each of which can be done independently in nk ways for k = 1, 2, .